Learning Interaction Kernels for Agent Systems on Riemannian Manifolds

Mauro Maggioni, Jason J Miller, Hongda Qiu, Ming Zhong
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:7290-7300, 2021.

Abstract

Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-maggioni21a, title = {Learning Interaction Kernels for Agent Systems on Riemannian Manifolds}, author = {Maggioni, Mauro and Miller, Jason J and Qiu, Hongda and Zhong, Ming}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {7290--7300}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/maggioni21a/maggioni21a.pdf}, url = {https://proceedings.mlr.press/v139/maggioni21a.html}, abstract = {Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.} }
Endnote
%0 Conference Paper %T Learning Interaction Kernels for Agent Systems on Riemannian Manifolds %A Mauro Maggioni %A Jason J Miller %A Hongda Qiu %A Ming Zhong %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-maggioni21a %I PMLR %P 7290--7300 %U https://proceedings.mlr.press/v139/maggioni21a.html %V 139 %X Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.
APA
Maggioni, M., Miller, J.J., Qiu, H. & Zhong, M.. (2021). Learning Interaction Kernels for Agent Systems on Riemannian Manifolds. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:7290-7300 Available from https://proceedings.mlr.press/v139/maggioni21a.html.

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